Canonical matrices for linear matrix problems

TL;DR

This paper develops a generalized canonical form for a broad class of matrix problems, extending Jordan normal form, and analyzes the structure of indecomposable matrices within this framework.

Contribution

It constructs Belitskii's algorithm for canonical reduction and characterizes the geometric structure of indecomposable matrices in the problem class.

Findings

C(m,n) is finite or consists of lines for all (m,n)

C(m,n) contains a plane for some (m,n)

The results unify classification of linear matrix problems

Abstract

We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskii's algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set C(m,n) of indecomposable canonical m-by-n matrices. Considering C(m,n) as a subset in the affine space of m-by-n matrices, we prove that either C(m,n) consists of a finite number of points and straight lines for every (m,n), or C(m,n) contains a 2-dimensional plane for a certain (m,n).